Let G be a connected graph. A subset S ⊆ V (G) is strong resolving dominating set of if and for every pair vertices u, v ∈ (G), there exists vertex w such that u IG[v, w] or IG[u, w]. The smallest cardinality called the domination number G. In this paper, we characterize sets in lexicographic product graphs determine corresponding number.

Given modal logics 1, 2, their lexicographic product 1 ⇤ 2 is a new logic whose frames are the Cartesian products of a 1-frame and a 2-frame, but with the new accessibility relations reminiscent of a lexicographic ordering. This article considers the lexicographic products of several modal logics with linear temporal logic (LTL) based on “next” and “always in the future”. We provide axiomatizat...

For a graph G and a positive integer k, a subset S of vertices of G is called a k-path vertex cover if every path of order k inG contains at least one vertex from S. The cardinality of a minimum k-path vertex cover is denoted by ψk(G). In this paper, we give some bounds and the exact values in special cases for ψk of the Cartesian, and lexicographic products of some graphs. Key–Words: Vertex co...

A description is given of finite permutation groups containing a cyclic regular subgroup. It is then applied to derive a classification of arc transitive circulants, completing the work dating from 1970’s. It is shown that a connected arc transitive circulant of order n is one of the following: a complete graph Kn , a lexicographic product [K̄b], a deleted lexicographic product [K̄b] − b , where ...

We will study the existence of different types of the Riesz Decomposition Property for the lexicographic product of two partially ordered groups. A special attention will be paid to the lexicographic product of the group of the integers with an arbitrary po-group. Then we apply these results to the study of n-perfect pseudo effect algebras. We show that the category of strong n-perfect pseudo-e...